If α and β are the zeroes of the quadratic polynomial f(x) = x2 – x – 2, find a polynomial whose
zeroes are 2α + 1 and 2β + 1.
Answers
Step-by-step explanation:
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EXPLANATION.
α, β are the zeroes of the quadratic polynomial.
⇒ f(x) = x² - x - 2.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ α + β = -(-1)/1 = 1. - - - - - (1).
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ αβ = (-2)/1 = - 2. - - - - - (2).
To find polynomial whose zeroes are,
⇒ 2α + 1 and 2β + 1.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ 2α + 1 + 2β + 1.
⇒ 2α + 2β + 2.
⇒ 2(α + β) + 2.
Put the value of α + β = 1 in the equation, we get.
⇒ 2(1) + 2 = 4.
⇒ α + β = 4. - - - - - (3).
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ (2α + 1)(2β + 1).
⇒ (2α)(2β) + (2α)(1) + (1)(2β) + (1)(1).
⇒ 4αβ + 2α + 2β + 1.
⇒ 4(αβ) + 2(α + β) + 1.
Put the value of αβ = - 2 and α + β = 1 in the equation, we get.
⇒ 4(-2) + 2(1) + 1.
⇒ - 8 + 2 + 1.
⇒ - 8 + 3 = - 5.
⇒ αβ = - 5. - - - - - (4).
As we know that,
Formula of the quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Put the value in the equation, we get.
⇒ x² - (4)x + (-5) = 0.
⇒ x² - 4x - 5 = 0.